A proposed capillary number dependent model for prediction of relative permeability in gas condensate reservoirs: a robust non-linear regression analysis Mehdi Mahdaviara, Abbas Helalizadeh

To cite this version:

Mehdi Mahdaviara, Abbas Helalizadeh. A proposed capillary number dependent model for prediction of relative permeability in gas condensate reservoirs: a robust non-linear regression analysis. Oil & Gas Science and Technology - Revue d’IFP Energies nouvelles, Institut Français du Pétrole, 2020, 75, pp.24. ￿10.2516/ogst/2020017￿. ￿hal-02557658￿

HAL Id: hal-02557658 https://hal.archives-ouvertes.fr/hal-02557658 Submitted on 28 Apr 2020

HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Oil & Gas Science and Technology – Rev. IFP Energies nouvelles 75, 24 (2020) Available online at: Ó M. Mahdaviara & A. Helalizadeh, published by IFP Energies nouvelles, 2020 ogst.ifpenergiesnouvelles.fr https://doi.org/10.2516/ogst/2020017

REGULAR ARTICLE

A proposed capillary number dependent model for prediction of relative permeability in gas condensate reservoirs: a robust non-linear regression analysis

Mehdi Mahdaviara1,* and Abbas Helalizadeh2

1 Department of Petroleum Engineering, Omidieh Branch, Islamic Azad University, 63731 93719 Omidieh, Iran 2 Department of Petroleum Engineering, Petroleum University of Technology (PUT), 61991 71183 Ahwaz, Iran

Received: 20 March 2019 / Accepted: 5 March 2020

Abstract. Well deliverability reduction as a result of liquid (condensate) build up in near well regions is an important deal in the development of gas condensate reservoirs. The relative permeability is an imperative fac- tor for characterization of the aforementioned problem. The dependence of relative permeability on the coupled effects of Interfacial Tension (IFT) and flow velocity (capillary number) together with phase saturation is well established in the literature. In gas condensate reservoirs, however, the influence of IFT and velocity on this parameter becomes more evident. The current paper aims to establish a new model for predicting the relative permeability of gas condensate reservoirs by employing the direct interpolation technique. To this end, the regression analysis was carried out using seven sets of literature published experimental data. The validity anal- ysis was executed by utilizing statistical parameters integrated with graphical descriptions. Furthermore, a comparison was carried out between the proposed model and some literature published empirical models. The results of the examination demonstrated that the new model outperformed other correlations from the standpoints of accuracy and reliability.

Nomenclature Rec Critical Reynolds number Sr Residual saturation Latin letters Sr Normalized residual saturation Swc Connate water saturation cd Drag coefficient v Velocity, ft/d Dm Mean grain size of the core f Weighting factor for linking kr to capillary Greek letters number e 1 f Weighting factor for linking sir to capillary b Forchheimer (non-Darcy) coefficient, ft number e Corey component k Absolute permeability, md l Viscosity, cp 3 kr Relative permeability q Density, lb /ft m kr End point relative permeability r Interfacial tension – krg/krc Gas condensate relative permeability ratio rb Base interfacial tension; highest measured n, m, a, b, interfacial tension and c Regression parameters (constants) / Porosity Nc Capillary number; the ratio of viscous to capillary forces Subscripts Ncb Base capillary number Re Reynolds number; the ratio of inertia to c Condensate phase viscous forces g Gas phase * Corresponding author: [email protected] I Immiscible condition

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 2 M. Mahdaviara and A. Helalizadeh: Oil & Gas Science and Technology – Rev. IFP Energies nouvelles 75, 24 (2020) i Phase indicator (gas or condensate) permeabilities of oil and water phases. A glass-etched water M Miscible condition wet pore network and two fluid systems with intermediate and strong wettabilities were employed for this purpose. Their analysis also revealed the vigorous correlation Abbreviations between the relative permeability and the capillary number. The two-phase flow structure (steady-state flow regime and CDC Capillary Desaturation Curve transient growth pattern) and the wettability type (interme- IFT Interfacial Tension diate and strong) were found as important factors influenc- ing the foregoing kr–Nc relationship. The transient kr of both phases was an increasing function of Nc in the case of strong wettability. The transient flow was associated with higher 1 Introduction oil relative permeability in comparison to the steady-state regime, whereas the steady-state water relative permeability The unique properties of the fluid in gas condensate reser- exceeded the transient one over the low Nc. Tsakiroglou voirs distinguishes its flow behavior from other oil and gas (2019) has also carried out a steady-state two-phase flow reservoirs. Condensate banking in near-well regions is a experiment to assess the relative permeability of the nitro- momentous phenomenon that affects the fluid flow in gen and brine in a homogeneous sand system. He reported porous media. This phenomenon occurs as a result of pres- a robust direct relationship between the gas and water sure dropping below reservoir fluids dew point pressure. relative permeabilities and the gas and water capillary The condensate banking plays an important role in well- numbers, respectively. The foregoing relationship was also deliverability because it brings about the accumulation of observed in the work of Ahmadi et al. (2019) in an attempt heavier components of the gas in pore space. In fact, a to investigate the effect of capillary number and reservoir gas condensate reservoir can chock on its most valuable heterogeneity on the location of injection/production wells components, namely condensate (Fan et al., 2005). There- using Computational Fluid Dynamics (CFD) approach. fore, a two-phase gas/condensate system (or three phases It is noteworthy that the aforementioned kr–Nc relationship when water is presented) emerges, in which diagnosing defines a Capillary Desaturation Curve (CDC). This param- the special manner of fluid flow/distribution in porous eter has been widely investigated in the context of IOR media (especially in the near well condition) is essential processes including water and chemical flooding (Delshad for production management. et al., 2009; Guo et al., 2015). The relative permeability (kr) indicates the competition Furthermore, the ratio of the inertia to viscous forces between the phases for flowing in porous media on a macro- (represented by Reynolds number) was found to be a scopic scale. This parameter mainly alters as a function of significant factor influencing the kr and Sr parameters fluid saturation. However, the dependence of relative per- (Henderson et al., 2000b). The Reynolds number plays a fl meability and residual saturation (Sr)ontheratioofviscous vital role in characterizing the governing ow regimes to capillary forces (denoted by the capillary number, Nc)in including laminar and turbulent. There are extraordinary a microscopic/pore scale is well established in the literature discrepancies between fluid flow in the porous media and (Ahmadi et al.,2019; Avraam and Payatakes, 1995; a conduit. The transition between the laminar and turbu- Delshad et al.,1986; Fulcher et al., 1985; Johannesen and lent flows is not detected obviously in the porous media Graue, 2007; Tsakiroglou, 2019; Tsakiroglou et al., 2007). (van Lopik et al.,2017). Furthermore, the flow regime cat- Avraam and Payatakes (1995) have represented a egorization is different in the stochastic nature of the hydro- pioneering work evaluating the relative permeability and carbon-bearing reservoirs. Dybbs and Edwards (1984) flow regime behaviors during a steady-state oil-water flow believe that the Reynolds number ranges of 1–10, 10–150, in a model pore space. They reported a robust relationship 150–250, 250–300, and greater than 300 correspond to the between the oil and water relative permeabilities and capil- inertial core establishment, non-linear steady laminar flow, lary number, the viscosity ratio, and the flow rate ratio. unsteady laminar flow, transition between unsteady and Both oil and water relative permeabilities increased by an highly unsteady laminar, and chaotic (turbulent) flow, increase in the capillary number. The foregoing relationship respectively (Al-Shaidi, 1997). Hence, the flow regime, as was related to the distribution of the oil ganglia/droplets, well as governing forces, serve as momentous factors affect- based on which the flow regime was classified into Large- ing the relative permeability of the reservoir. Ganglion Dynamics (LGD), Small-Ganglion Dynamics Typically, the distribution of the phases in pore spaces (SGD), Drop-Traffic Flow (DTF), and Connected Pathway is dominated by capillary forces (Blom and Hagoort, Flow (CPF). The formation of connected oil pathways was 1998). The magnitude of the capillary forces is determined corresponded to the largest capillary numbers and relative by Interfacial Tension (IFT), wettability, and pore geome- permeabilities. Fulcher et al. (1985) have implemented a try (Delshad et al.,1986). It is well recognized that a steady-state relative permeability measurement to investi- reduction in interfacial tension (the capillary number gate the effect of capillary number on the relative permeabil- increment) will entail an increase in relative permeability ity curves. They also observed a close relationship between as well as fluid residual saturations reduction (Sr) these parameters, especially in the case of low IFTs. (Amaefule and Handy, 1982; Chukwudeme et al., 2014). Tsakiroglou et al. (2007) have utilized the history match- In gas condensate reservoirs, however, the viscous forces ing of transient flow experiments to assess the relative may be of the same order of magnitude as the capillary M. Mahdaviara and A. Helalizadeh: Oil & Gas Science and Technology – Rev. IFP Energies nouvelles 75, 24 (2020) 3 forces (Blom and Hagoort, 1998). The immensity of the Blom and Hagoort (1998) divided the foregoing correla- viscous forces is set by the fluid viscosity, flow velocity, tions into two major categories according to the manner of and flow path length (Fulcher et al., 1985). It has been including the capillary number: widely accepted that the foregoing connection between the IFT and kr/Sr becomes momentous when the IFT is 1. Utilizing Corey function, in which its coefficients were lower than its critical (base) value (IFTb)(Asar and Handy, linked to Nc (Eq. (1))(App and Burger, 2009; Blom 1988; Kalla et al., 2014; Longeron, 1980). Additionally, and Hagoort, 1998; Fulcher et al., 1985): many researchers have been confirmed experimentally that relative permeability of the gas condensate fluids (in e ðÞ s s ðÞN i N c particular for the gas phase) increase with the velocity at k ðÞ¼N ; s k ðÞN i ri c ; ð1Þ ri c i ri c ðÞ low/moderate flow velocities (Henderson et al.,1997, 1 sri N c swc 2000a; Jamiolahmady et al., 2008; Mott et al.,1999). Con- versely, relative permeability reduction with an increase in where i refers to phase type (gas or condensate), kri velocity at high velocities is also well established. This phe- (Nc,si) stands for saturation/capillary number dependent e nomenon takes place as a result of non-Darcy flow in the relative permeability; and kri (Nc), Sri (Nc) and i (Nc) are capillary number (or IFT) dependent Corey coefficients. vicinity of the wellbore (in particular for the gas phase). The positive effect of IFT together with low/moderate The kri (Nc) denotes the endpoint relative permeability, e velocities on relative permeability has been termed Sri (Nc) stands for residual saturation and i (Nc) is the fi “coupling effect” (Avraam and Payatakes, 1995; Henderson Corey component, which xes the curvature of the et al., 2000a; Jamiolahmady et al.,2008, 2009); and the relative permeability function. negative effect of high velocities on this parameter is known as “inertia effect”. Accordingly, the interaction of coupling 2. Interpolation between miscible and immiscible rela- and inertia effects determines the shape and the value of tive permeability curves (Eq. (2))(Amaefule and relative permeability curves (Henderson et al., 2000b). Handy, 1982; Betté et al., 1991; Henderson et al., Taking into account the coupling and inertia effects, as 2000a; Jamiolahmady et al., 2009; Whitson and well as saturation, relies at the heart of the relative perme- Fevang, 1997): ability measurement/calculation in gas condensate ðÞ¼; ðÞ þ ½ ðÞ ; ð Þ reservoirs. Since the relative permeability experimentally K ri N c si fi N c K riI 1 fi N c K riM 2 measurement process is expensive and time-consuming, correlations describing relative permeability are extremely where kriI and kriM are immiscible (or rock) and miscible useful in the absence of laboratory-measured data. Accord- (or straight line) relative permeabilities, respectively. All ingly, there are several literature-published correlations for the intermediate relative permeability points are situated estimation of relative permeability in conventional reser- between these two limits. The fi (Nc), an Nc dependent voirs containing immiscible fluids (Brooks and Corey, function, determines the proximity of the relative perme- 1964; Lomeland et al., 2005; Stone, 1973). Additionally, ability curve to the aforesaid limits. The value of this different researchers have been attempted to link the weighting factor changes between 0 for fully miscible relative permeability of immiscible fluids to the capillary (high Nc), and 1 (low Nc) for completely immiscible number and saturation in accordance to the steady-state mixtures. ’ two-phase flow experiments (Lenormand et al.,1988; To the best of authors knowledge, Coats (1980) has fi Payatakes, 1982; Tsakiroglou et al., 2007, 2015). employed the interpolation technique for the rst time to In the case of gas condensate and near-critical fluids, link the kr to the IFT. This simple intuitive approach (with- various empirical models have been developed with respect out any explicit physical meaning) has been grabbed the to the fact that the coupled effects of IFT and velocity attention of the majority of researchers (Amaefule and momentously dominate the relative permeability curves of Handy, 1982; Betté et al.,1991; Henderson et al.,2000a; the foregoing fluids (Coats, 1980; Jamiolahmady et al., Jamiolahmady et al.,2009; Whitson and Fevang, 1997). 2009; Pope et al., 2000; Whitson and Fevang, 1997). The Different authors have proposed various weighting factors for linking the kr/Sr to the Nc. Furthermore, Whitson and majority of the aforesaid correlations employ the Nc for taking into account the coupling effect (Al-Shaidi, 1997; Fevang (1997) interpolated the kr points as a function of Whitson and Fevang, 1997). However, since the positive non-wet to wet relative permeability ratio (krg/krc)asan – effect of velocity on relative permeability was not perceived alternative to saturation. Accordingly, the Sr Nc relation- by early authors, they proposed models directly according ship was ignored. However, this technique was only applica- to the IFT (Betté et al.,1991; Coats, 1980; Nghiem et al., ble to near-wellbore regions, where both gas and condensate fl 1981). Additionally, Jamiolahmady et al. (2006) employed phases are owing. both IFT and N parameters simultaneously, because they For considering the negative effect of inertia, on the c other hand, the Forchheimer equation (Eq. (3))hasbeen believed that presence of IFT in the denominator of Nc equation does not enough to express the dependency on employed as follows (App and Burger, 2009; Jamiolahmady IFT. Furthermore, Pope et al. (2000) utilized trapping et al.,2009; Whitson et al.,1999): number (a generalization of the capillary and bond l dP v 2 numbers) as an alternative to the Nc in order to account ¼ þ bqv ; ð3Þ for the gravitational forces. dL k 4 M. Mahdaviara and A. Helalizadeh: Oil & Gas Science and Technology – Rev. IFP Energies nouvelles 75, 24 (2020) where b stands for Forchheimer (or non-Darcy) coefficient miscible relative permeability, which alters as a function with L1 dimension, k is absolute permeability, and q, v, of capillary number dependence residual saturation as and l denote density, velocity, and viscosity. well as phase saturation. Although the aforementioned correlations are worth- while in the absence of experimental data, they are associ- 2. The relative permeability curves as functions of non- ated with shortcomings in various aspects. The majority wet to wet relative permeability ratio (krg/krc) instead of these models suffer from a large number of tuning param- of saturation: eters (constants), in which the setting value of each param- eter is divergent for various cases. Accordingly, designating krg krg krg K ri N ci; ¼ fiðÞN ci K riI þ ½1 fiðÞN ci K riM ; the expedient values for aforesaid regression parameters is a krc krc krc tough duty. The problem becomes even more pronounced ð5Þ when the accuracy and reliability of each model are not fl unique for various rock and uid systems. Each model has where kriI(krg/krc) and kriM(krg/krc) are immiscible and been correlated by implementing a special experiment on miscible relative permeabilities, respectively. The afore- a limited number of rock and fluid systems and experimen- mentioned parameters are functions of non-wet to wet tal conditions. Accordingly, there is uncertainty for choos- relative permeability ratio (krg/krc) instead of saturation. ing the most appropriate model for estimation of relative permeability in a particular gas condensate reservoir. 2.1.1 The weighting factor In order to alleviate these limitations in the current study, the authors attempted to develop a new reliable A vast variety of data points from seven sets of literature published datasets (Asar and Handy, 1988; Calisgan and model for the prediction of kr in gas condensate reservoirs Akin, 2008; Chen et al.,1995; Haniff and Ali, 1990; as a function of Nc as well as saturation. The authors aimed advantages of simplicity and the least number of tuning Henderson et al.,1997, 1988; Longeron, 1980) were parameters (constants)/variables integrated with averagely employed to develop a precise and reliable weighting func- high reliability for various rock and fluid systems. To this tion. To this end, regression analysis was carried out using end, regression analysis was carried out by employing seven Gauss–Newton and Levenberg–Marquardt algorithms. This sets of literature experimentally data (Asar and Handy, examination led to the following logistic weighting factor: 1988; Calisgan and Akin, 2008; Chen et al., 1995; Haniff Ai and Ali, 1990; Henderson et al.,1997, 1988; Longeron, fiðÞ¼N ci ; ð6Þ N ci 1 þ Q exp Bi 1980). The Capillary and Reynolds numbers were assessed i N cib to recognize the governing flow regime/forces over the utilized datasets. The performance of the developed correla- where i stands for phase type (namely gas or condensate), tions was investigated using statistical parameters of Root Nci denotes the capillary number, and Ncib represents the Mean Square Error (RMSE) and determination coefficient base capillary number, below which the relative perme- (R2). The validity of the aforesaid empirical functions was ability of the gas and condensate becomes independent evaluated by implementing a comparison with other from the capillary number (namely, the value of the literature proposed models (Al-Shaidi, 1997; Coats, 1980; capillary number at immiscible condition; i.e., at highest Jamiolahmady et al.,2009; Nghiem et al.,1981; Whitson IFT and lowest velocity). The entities Ai, Qi,andBi are and Fevang, 1997). To provide more investigation regard- the tuning parameters (constants) of the equation. Default fi ing the accuracy of the developed model, curve tting anal- values for the entities Ai and Qi are 2 and 1, respectively. ysis was carried out by utilizing a set of previously Given these values, Bi changes between 0 and 0.5 for both published data (Blom et al.,1997) which had been excluded gas and condensate phases. The value of the Bi parameter from the regression procedure. is relatively unique for gas and condensate phases (Bg Bc). Furthermore, a correlation was developed for linking the tuning parameter of Bi to the multiplication 2 Model description of porosity and permeability as follows: m s ni 2.1 Applying the coupling effect B ¼ a i i ; ð7Þ i i ð; kÞci Two different structures of the direct interpolation were uti- lized for linking the Nc (coupling effect) to the relative where, the entities u and k are porosity and absolute per- permeability: meability (md), respectively. The default values of mi and ci for both phases are 3 and 1, respectively. The value of ni 1. The relative permeability curves as functions of phase is 0.63 and 1.77 for gas and condensate phases, saturation: respectively. N K ðÞ¼N ; s f ðÞN K ðÞs ; s For small c, capillary forces dominate and tradi- ri ci i i ci riI i ir tional (immiscible) relative permeability behavior is found. þ ½1 fiðÞN ci K riMðÞsi; sirðÞN ci ; ð4Þ Consequently, for large Nc, viscous forces dominate and relative permeabilities tend to approach straight lines or where KriI (si,sir) denotes saturation dependent immisci- miscible like behavior (Whitson et al.,1999). Hence, the ble relative permeability, and KriM (si,sir (Nci)) stands for value of the proposed weighting factor of equation (6) alters M. Mahdaviara and A. Helalizadeh: Oil & Gas Science and Technology – Rev. IFP Energies nouvelles 75, 24 (2020) 5

3 between 0 and 1 for thoroughly miscible (high Nc)and where the entity qi is density in lbm/ft , vi denotes the completely immiscible (small Nc) conditions, respectively. velocity in ft/d, and li is viscosity in cp. Thereby, for a given saturation or krg/krc point, the gas 2.1.2 The immiscible relative permeability condensate relative permeability can be estimated accord- ingtothefollowingsteps: Conventional experimental measurements or traditional correlations such as Brooks and Corey (1964) and Stone 1. Calculate the immiscible relative permeability using (1973) can be employed to achieve the immiscible relative experimental measurements or traditional correlations permeability curves. such as Brooks and Corey (1964) and Stone (1973). 2. Link the residual saturation to the capillary number by 2.1.3 The miscible relative permeability utilizing equation (10) integrated with equation (6). The modified relative permeability correlation of equation 3. Calculate the miscible relative permeability by (1) is proposed for the miscible state: employing equations (8) or (9). 4. Measure the value of the weighting factor using s s ðÞN s s ðN Þ ¼ i ir ci ¼ i ir ci ; ð Þ equation (6). kriM ðÞ ð Þ 8 1 swc sir N ci 1 sir N ci 5. Estimate the capillary number dependent relative per- meability of the given saturation and/or krg/krc point where swc stands for connate water saturation, sir (Nc)is by manipulating equations (4) or (5). capillary number dependent residual saturation and sir 6. Utilize equation (11) for applying the influence of (Nc) denotes normalized capillary number dependent inertia. residual saturation. This is noteworthy that the following expression can be The flowchart of Figure 1 represents a visual description utilized for obtaining the miscible relative permeability of the sequence of relative permeability prediction for gas curves directly as a function of krg/krc: condensate reservoirs. 1 s k ¼ wc ; ð9Þ rgM þð = Þ1 1 krg krc 3 Results and discussions where (1) and (+1) indexes must be used for calculation of the gas and condensate miscible relative permeabilities, In the current study, a cumulative of seven datasets were respectively. collected from the literature (Asar and Handy, 1988; The residual saturation was related to the capillary Calisgan and Akin, 2008; Chen et al.,1995; Haniff and number in the same manner of other literature published Ali, 1990; Henderson et al.,1997, 1988; Longeron, 1980) models: for developing a model for prediction of relative permeability in gas condensate reservoirs. A description of the aforesaid ðÞ¼e ðÞ; ð Þ sir N c f i N ci sirI 10 datasets including rock and fluid properties as well as the experiment methods (namely, steady-state and unsteady- where s is the residual saturation at immiscible condi- irI ~ state) is represented in Table 1. As can be seen in the table, tion, and f i (Nci) denotes the weighting factor utilized these datasets encompass various rock and fluid systems, for linking sirI to the Nc. Regression analysis was executed experimental methods, and the widespread points of gas once again in order to obtain a reliable expression for the and condensate relative permeabilities as a function of satu- aforesaid weighting function. The results of the analysis ration, IFT, and capillary number. Accordingly, the devel- demonstrated that equation (6) is also reliable for connect- oped model becomes more applicable to predict the ing the capillary number to the residual saturation. relative permeability of different gas condensate reservoirs. Equation (6) makes use of the fact that residual saturation The above-mentioned datasets reported the dependency tends toward zero by an increase in the capillary number. of relative permeability to the velocity and IFT without At the end, it should be worth pointing out that the addressing the flow nature/regime of the experiments. effect of inertia can be considered by employing the Darcy Accordingly, the pore-scale dimensionless numbers of equation integrated with Forchheimer expression as follows Capillary and Reynolds were employed to assess the (Al-Shaidi, 1997): regimes/forces governing the fluid flow in each dataset. 1 For this purpose, the pore-scale Capillary number was k ðÞ¼N ; s ; b ; ð11Þ ri ci i i 1 þ : 16 b determined by utilizing the following literature correlation ðÞ; 1 83 10 i k Re kri N ci si (Foster, 1973): l v where kri (Nci,si, bi) is coupling, saturation, and inertia i i N c ¼ : ð13Þ dependent relative permeability, k denotes the absolute ri; permeability in md, bi stands for Forchheimer coefficient in ft1, and Re is the Reynolds number which defines as Furthermore, the pore-scale Reynolds number was the ratio of inertial to viscous forces as follows: calculated as follows (Bear, 2013): q v q v D ¼ i i ; ð Þ Re ¼ i i mi ; ð14Þ Re l 12 l i i 6 M. Mahdaviara and A. Helalizadeh: Oil & Gas Science and Technology – Rev. IFP Energies nouvelles 75, 24 (2020)

Start

Measure the weighting Eq. (6) factor

Calculate the Calculate the Exp. Exp. immiscible kr as a immiscible k as a /Corr. /Corr. r function of krg/krc function of saturation

Eqs. (6) Link the residual and saturation to the (10) capillary number

Calculate the miscible Calculate the miscible kr as a function of Eq. (9) Eq. (8) kr as a function of krg/krc saturation

Estimate the capillary Estimate the capillary Eq. (5) number dependent kr as Eq. (4) number dependent kr a function of krg/krc as a function of saturation

Eq. Apply the effect of (11) inertia

End

Fig. 1. Flowchart of the relative permeability estimation in gas condensate reservoirs.

1=3 where Dm stands for the mean grain size of the core. The 50:8;ð1 ;Þ Re ¼ ; ð16Þ following equation was utilized to calculate the foregoing c ð ð ;Þ1=3Þ parameter for each dataset (Urumović and Urumović, cd 1 1 2016): rffiffiffiffiffiffiffiffiffiffi where, cd represents the drag coefficient, which corre- ð ;Þ sponds to the value of 1.9 in the current study. ¼ 1 180k: ð Þ Dm ; ; 15 The values of the approximated Capillary and Reynolds numbers together with the critical Reynolds number are The critical Reynolds number, above which the inertia demonstrated in Table 2 for various experimental condi- effect is dominant, was quantified as follows (Du Plessis tions. An expeditious examination of the Capillary number and Woudberg, 2008): values indicates that the capillary forces are dominant M. Mahdaviara and A. Helalizadeh: Oil & Gas Science and Technology – Rev. IFP Energies nouvelles 75, 24 (2020) 7

Table 1. Summary of the experimental conditions of the seven literature datasets utilized in this study. Dataset Dataset 1 Dataset 2 Dataset 3 Dataset 4 Dataset 5 Dataset 6 Dataset 7 (Longeron, (Asar and (Haniff and (Chen et al., (Henderson (Henderson (Calisgan and 1980) Handy, 1988) Ali, 1990) 1995) et al., 1997) et al., 1988) Akin 2008) Experiment U.S.S S.S S.S S.S S.S S.S S.S method Rock type Fontainebleau Berea sand Spynie sand North Sea Berea sand Berea sand Carbonate sand stone stone stone gas stone stone condensate Porosity (%) 9.9 20 22 17.4 18.2 19.8 15.8* Absolute 83 193 23 73.39 92 92 18.56 permeability (md) Irreducible 0 0 0 0.209 26.4 26.4 0 water (%) Fluid C1–nC7 C1–C3 C1–C3 North Sea C1–C3–nC5– C1–nC4 C1–nC6 composition gas nC10–nC16 condensate Temperature 71.1 21 31.7 121.1 – 37 – (°C) IFT (mN/m) 0.001–12.6 0.03–0.83 0.001–0.05 0.03–0.35 0.05–0.4 0.14–0.9 0.01–0.39

Nc (e-6) 0.05–1100 0.443–148 5.56–947 5.36–42.9 9.4–36 0.28–14 0.124–8.62 – ( ): means that datap wereffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi not reported; (*): the porosity value was predicted by utilizing correlation proposed by ;¼ 4:5 : 2 (Timur, 1968)( 0 136kswr). S.S: steady state, U.S.S: unsteady state. factors governing the fluid flow in the pore scale. However, an appropriate mathematical expression for the weighting the influence of the viscous force increases when the Nc factor. The analysis led to the capillary dependent logistic towards unity by an increase in velocity as well as a function of equation (6). This function represents the decrease in IFT. Additionally, the evaluation of the interaction between the capillary and viscous forces regard- Reynolds number values reveals the governance of the vis- less of the inertia effect due to the governance of the steady cous forces in comparison to the inertia. The calculated Re laminar flow regime in all employed datasets. In low values ranged from 2.93E-06 to 8.72E-04 for all datasets. capillary numbers (high IFT/low velocity), in which the The critical Reynolds numbers correspond to the 74.8553, capillary forces are entirely dominant, the weighting factor 69.2509, 68.1219, 69.3634, 69.3634, 69.3634, and 71.6003 corresponds to unity. The weighting factor towards zero by values for datasets 1–7, respectively. Although the Re val- increasing the viscous force that arises from an increase in ues also increase by an increase in velocity, they are remark- capillary number (low IFT/moderate velocity). As a conse- ably lower than their critical value in all employed quence, the gas and condensate relative permeability literature datasets. Accordingly, the negative effect of iner- curves improve and tend towards the unit slope line. Two tia was not detected over the relative permeability points discrete reasons emerged from this (Blom and Hagoort, utilized in this study, designating that the flow was within 1998). The viscous force increment may allow a short-cut the laminar regime. Additionally, the fluid flow was chiefly in pathways that were blocked under capillary force gover- governed by the capillary and viscous forces, which sup- nance. Furthermore, an increase in viscous force may ports the aforementioned literature findings. deploy motionless fluids such as isolated blobs, namely For fulfilling the scope of the current study, the direct the sources of the residual saturation. linear interpolation was utilized to predict the relative per- Accordingly, the identical weighting factor was meability as a function of saturation (Eq. (5)) and non-wet employed for linking the residual saturations of the immis- to wet relative permeability ratio (Eq. (6)). The foregoing cible and miscible conditions. As previously stated, the correlations use a capillary number dependent weighting residual saturation of both phases towards zero by an factor to predict a certain relative permeably point through enhancement in viscous forces. Subsequently, the weighting its corresponding immiscible and miscible relative perme- factor towards zero by the capillary number increment. abilities. A comprehensive regression analysis was carried The values of the tuning parameters of equation (6) out by utilizing the above-described datasets to achieve for relative permeability and residual saturation are 8 M. Mahdaviara and A. Helalizadeh: Oil & Gas Science and Technology – Rev. IFP Energies nouvelles 75, 24 (2020)

Table 2. The calculated dimensionless numbers over various IFT and velocity values.

Dataset Experiment method IFT v (m/s) Nc Re Rec Dataset 1 Unsteady state 0.065 5.61E-05 1.92E-04 8.57E-04 74.85532 (Longeron, 1980) 0.040 5.56E-05 3.84E-04 8.49E-04 0.020 5.03E-05 6.57E-03 7.68E-04 0.001 5.71E-05 1.11E-02 8.72E-04 Dataset 2 (Asar Steady state 0.830 3.61E-05 2.22E-06 1.90E-05 69.25087 and Handy, 1988) 0.430 3.82E-04 4.47E-05 2.01E-04 0.180 4.13E-04 1.16E-04 2.17E-04 0.030 4.41E-04 7.40E-04 2.32E-04 Dataset 3 (Haniff Steady state 0.200 1.07E-04 2.53E-05 1.53E-05 68.12191 and Ali, 1990) 0.100 8.55E-05 4.04E-05 1.22E-05 0.050 8.71E-05 8.23E-05 1.25E-05 0.010 8.96E-05 4.23E-04 1.28E-05 0.001 9.12E-05 4.30E-03 1.3E-05 Dataset 4 (Chen Steady state 0.034-0.349 4.94E-06 3.08E-05 1.27E-04 70.70861 et al.,1995) 9.89E-06 6.16E-05 2.55E-04 1.98E-05 1.23E-04 5.09E-04 3.95E-05 2.46E-04 1.02E-03 Dataset 5 Steady state 0.400 1.02E-05 4.55E-06 2.93E-06 69.36340 (Henderson et al., 9.84E-05 4.55E-05 2.83E-05 1997) 4.07E-04 1.82E-04 1.17E-04 0.050 1.02E-05 4.75E-05 2.93E-06 9.84E-05 4.75E-04 2.83E-05 4.07E-04 1.92E-03 1.17E-04 Dataset 6 Steady state 0.900 1.07E-04 1.41E-06 3.08E-05 69.36340 (Henderson et al., 2.14E-04 2.83E-06 6.16E-05 1988) 4.28E-04 5.66E-06 1.23E-04 8.56E-04 1.13E-05 2.46E-04 0.140 1.07E-04 9.09E-06 3.08E-05 2.14E-04 1.82E-05 6.16E-05 4.28E-04 3.64E-05 1.23E-04 8.56E-04 7.07E-05 2.46E-04 Dataset 7 Steady state 0.3933 2.6556E-06 5.46E-05 1.03E-05 71.60033 (Calisgan and 1.3278E-06 5.46E-05 1.03E-05 Akin, 2008) 0.0590 2.4416E-06 5.46E-05 1.03E-05 1.2192E-06 5.46E-05 1.03E-05 0.0100 2.3469E-06 5.46E-05 1.03E-05 1.7588E-06 5.46E-05 1.03E-05 1.1734E-06 5.46E-05 1.03E-05

represented in Tables 3 and 4, respectively. As can be seen relationship with saturation and multiplication of porosity in Table 3, the constant parameters of the logistic function and permeability was evaluated using regression analysis. are situated in the limited ranges. According to the author’s The results of the analysis are represented in Figures 2 observations, choosing the default values of A = 2 and and 3 for gas and condensate phases, respectively. Two dif- Q = 1 is not associated with significant errors. Thereby, ferent graphical plots including contour and surface plots the value of Bi will be situated in the range of 0–0.5 for both are represented to demonstrate the results more phases. Additionally, for obtaining the exact value of Bi,its descriptively. As it is evident, the tuning parameter of Bi M. Mahdaviara and A. Helalizadeh: Oil & Gas Science and Technology – Rev. IFP Energies nouvelles 75, 24 (2020) 9

Table 3. Values of the regression parameters (constants) as well as RMSE and R2 entities for the weighting factor of equation (6) (kr–Nc) in various datasets used in the current study. AQ BRMSE R2 GCG CGC GCGC Dataset 1 0.83 2.000 2.130E-5 1.406 0.290 12.450E-5 0.1319 0.1378 0.8827 0.8390 Dataset 2 1.028 1.000 0.025 0.016 0.100 0.099 0.1100 0.0376 0.9447 0.9927 Dataset 3 2.000 2.000 0.570 0.463 0.690 0.790 0.1132 0.0714 0.9147 0.9710 Dataset 4 1.100 – 0.500 – 0.780 – 0.0830 – 0.9653 – Dataset 5 2.000 2.000 1.310 1.086 0.013 0.009 0.0958 0.0651 0.9139 0.9652 Dataset 6 2.000 2.000 1.000 1.000 0.073 0.065 0.0560 0.0721 0.9704 0.9570 Dataset 7 2.000 – 1.000 – 0.053 – 0.0787 – 0.9559 – (–): means that the data were not reported.

Table 4. Values of the regression parameters (constants) as well as RMSE and R2 entities for the weighting factor of equation (6) (Sr–Nc) in various datasets used in the current study. AQ B RMSE R2 GCGCG C GCGC Dataset 1 – 0.840 – 10.000 – 1527.820 – 0.1198 – 0.9228 Dataset 2 1.000 1.000 10.899 2.967 48.505 14.643 0.0252 0.0301 0.9987 0.9949 Dataset 3 1.198 1.000 8.846 16.857 3.785 6.369 0.0390 0.0415 0.9946 0.9945 Dataset 4 1.000 – 0.109 – 5.746 – 0.0003 – 0.9991 – Dataset 5 0.938 1.000 1.823 0.915 128.026 63.131 0.0646 0.0502 0.9653 0.9593 Dataset 6 0.970 1.000 0.670 0.515 24.448 6.231 0.0382 0.0010 0.9300 0.9999 Dataset 7 1.000 – 0.880 – 0.109 – 0.0342 – 0.9885 – (–): means that the data were not reported.

(a) (b)

Fig. 2. Employing the (a) surface and (b) contour plots in order to investigate the relationship between Bg (regression parameter of the Eq. (6) for gas), gas saturation and multiplication of the porosity and the permeability (md). vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi increases by saturation and decreases by the multiplication u u XN of porosity and permeability. t 1 2 RMSE ¼ ðf ; f ; Þ ; ð17Þ N exp i pred i 3.1 Validity analysis of the proposed weighting factor i¼1 PN 2 Accuracy and reliability of the proposed logistic weighting ðfexp; i fpred; iÞ ¼ factor were surveyed by employing the Root Mean Square R2 ¼ 1 i 1 ; ð18Þ 2 PN 2 Error (RMSE) and R-squared (R ) statistical parameters. ðfexp; i fexp; iÞ The aforesaid parameters are defined as follows: i¼1 10 M. Mahdaviara and A. Helalizadeh: Oil & Gas Science and Technology – Rev. IFP Energies nouvelles 75, 24 (2020)

(a) (b)

Fig. 3. Employing the (a) surface and (b) contour plots in order to investigate the relationship between Bc (regression parameter of the Eq. (6) for condensate), condensate saturation and multiplication of the porosity and the permeability (md) for the condensate phase. where, N stands for the number of data points, and the sub- equations due to providing further assessment concerning scripts of exp and pred demonstrate the experimental and the new model. In this regard, curve fitting analysis was car- predicted values of the weighting factor, respectively. ried out using seven sets of data employed in the current It is noteworthy that the R-squared entity is not an study. Root Mean Square Error (RMSE) and R-squared appropriate statistical parameter for investigating the accu- (R2) statistical parameters were used for determining the racy of the non-linear regressions; however, it was reported accuracy and the fit goodness of the calculated values on habitually in the current study. the experimental points. The values of the statistical parameters of RMSE and 2 R for equation (6) are represented in Tables 3 and 4 for rel- 3.1.1.1 The weighting factor of equation (6) for relative permeability ative permeability and residual saturation, respectively. As The proposed weighting factor of equation (6) was com- shown in the tables, this equation represents averagely fair pared with five literature correlations represented in enough accuracy for linking the relative permeability and Table 5. As shown in the table, the number of variables residual saturation to the capillary number. Furthermore, of each equation is different from others. The proposed a visual representation of the results of the regression anal- weighting factor of equation (6) as well as equations f1,f2 ysis for developing the weighting factor of equation (6) are and f are only capillary number dependent. However, – 4 shown in Figures 4 7 for relative permeability/residual sat- equations f3 and f5 are functions of saturation and IFT uration and gas/condensate phases. The solid points indi- (r), respectively, together with the capillary number. cate various experimental relative permeability data Results of the comparison are shown in Figures 8 and 9 points, and the solid lines demonstrate the calculated trend for gas and condensate phases, respectively. As can be seen, using the new correlation. It should be noted that the actual the new weighting factor demonstrates averagely more pre- experimental points are capillary number dependent. The cision in comparison with other single/two variable equa- weighting factors decrease from 1 to 0 by increasing the tions. Moreover, this logistic function provides enough capillary number parameter. Furthermore, the experimen- fitting accuracy consistently in all of the datasets. However, tal points corresponding to the weighting factor of relative other literature correlations do not indicate uniformly permeability are saturation dependent as well as the depen- acceptable agreement with the actual experimental points, dency on the capillary number. However, for the sake of because they were developed only using a limited number simplicity, the developed weighting factor in this study of experimental results. Hence, the developed model in this was linked only to the capillary number parameter. In fact, study provides more reliability for predicting the relative the maximum achievable accuracy was sacrificed for sim- permeability of a gas condensate reservoir, regardless of plicity. On the other hand, the decreasing trend of weight- the type of rock and fluid system. ing factors with the capillary number entity is not unique for all of the datasets. Nevertheless, the proposed weighting factor indicates the ability to fit onto different shapes such 3.1.1.2 The weighting factor of equation (6) for residual saturation The curve fitting procedure was repeated to investigate the as S-shaped and exponential curves. This is another affir- ability of aforesaid expressions for linking the capillary mation for the accuracy and reliability of the developed number to the residual saturation. The results of the weighting factor in this study. comparison are shown in Figures 10 and 11 for gas and 3.1.1 Comparison with other literature models condensate phases, respectively. As it is evident, the pro- posed weighting factor of equation (6) indicates more A comparison was conducted between the proposed accuracy for both gas and condensate phase than other weighting factors of equation (6) and the literature proposed literature correlations. It should be noted that the f4 M. Mahdaviara and A. Helalizadeh: Oil & Gas Science and Technology – Rev. IFP Energies nouvelles 75, 24 (2020) 11

(a) (b)

(c) (d)

(e) (f)

(g)

Fig. 4. Investigating the accuracy/precision of the fg (the proposed weighting factor of Eq. (6) (kr–Nc) for the gas phase) for fitting on seven employed literature datasets, namely (a) Dataset 1, (b) Dataset 2, (c) Dataset 3, (d) Dataset 4, (e) Dataset 5, (f) Dataset 6, and (g) Dataset 7. 12 M. Mahdaviara and A. Helalizadeh: Oil & Gas Science and Technology – Rev. IFP Energies nouvelles 75, 24 (2020)

(a) (b)

(c) (d)

(e)

Fig. 5. Investigating the accuracy/precision of the fc (the proposed weighting factor of Eq. (6) (kr–Nc) for the condensate phase) for fitting on five employed literature datasets, namely (a) Dataset 1, (b) Dataset 2, (c) Dataset 3, (d) Dataset 5, and (e) Dataset 6. It should be noted that Datasets 4 and 7 were excluded from the analysis because they were not available for the condensate phase.

weighting factor was not successful for linking the residual 3.1.2 Estimation of the relative permeability saturation to the capillary number; therefore, the values of the statistical parameters of this equation were not At the end, the proposed model in this study was utilized reported in the figures. for the estimation of relative permeability in a gas conden- M. Mahdaviara and A. Helalizadeh: Oil & Gas Science and Technology – Rev. IFP Energies nouvelles 75, 24 (2020) 13

(a) (b)

(c) (d)

(e) (f)

~ Fig. 6. Investigating the accuracy/precision of the f g (the proposed weighting factor of Eq. (6) (Sr–Nc) for the gas phase) for fitting on six employed literature datasets, namely (a) Dataset 2, (b) Dataset 3, (c) Dataset 4, (d) Dataset 5, (e) Dataset 6, and (f) Dataset 7. It should be noted that Dataset 1 was excluded from the analysis because it was not reported for this purpose. sate reservoir. For this purpose, a recent experimental non-wetting phase) and Methanol (as wetting phase) were dataset was adopted from the literature (Blom et al., used as the fluid system of the experiment. The Capillary 1997). These data points were achieved by executing an and Reynolds numbers were calculated at the pore level unsteady-state experiment. The rock system was a glass to recognize the governing flow regime in this dataset. porous media with porosity and permeability of 0.36 and The Capillary number increased from 7.42E-4 to 4.25E-2 972.72 md, respectively. Additionally, the Hexane (as by an increase in viscous force dominance. The Reynolds 14 M. Mahdaviara and A. Helalizadeh: Oil & Gas Science and Technology – Rev. IFP Energies nouvelles 75, 24 (2020)

(a) (b)

(c) (d)

(e)

~ Fig. 7. Investigating the accuracy/precision of the f g (the proposed weighting factor of Eq. (6) (Sr–Nc) for the condensate phase) for fitting on five employed literature datasets, namely (a) Dataset 1, (b) Dataset 2, (c) Dataset 3, (d) Dataset 5, and (e) Dataset 6. It should be noted that Datasets 4 and 7 were excluded from the analysis because they were not available for the condensate phase.

Table 5. Five literature published correlations employed for the comparison process.

Authors fi Equation no. 1 n Coats (1980) N cib i f1 N ci Ncib ni Nghiem et al. (1981) 1 e Nci f2 n2i n1i si Al-Shaidi (1997) N cib f3 N ci

1 Whitson and Fevang (1997) n2i f4 ðn1i N ciÞ þ1 hi r Nci 0 1þn1i log ðÞr hi Ncibnohi Jamiolahmady et al. (2009) 2 f5 r r Nci 0 Nci 0 1þn1i log ðÞr þn2i log ðÞr Ncib Ncib M. Mahdaviara and A. Helalizadeh: Oil & Gas Science and Technology – Rev. IFP Energies nouvelles 75, 24 (2020) 15

(a)

(b)

Fig. 8. Comparing the accuracy/precision of the new weighting factor of equation (6) (kr–Nc) with five literature published correlations of f1–f5 (Tab. 5) for the gas phase. (a) The RMSE and (b) R2 entities were employed for determining the fit goodness of each function on seven experimental datasets of Table 1.

number alters in a range from 0.0019 to 0.0025, which is sets of inputs, namely Nc and saturation as well as Nc and much less than the attributed critical Reynolds number of krg/krc. The experimental relative permeability curves corre- 60.008. Hence, the entirely laminar flow regime governs sponding to capillary numbers of 2.67E-4and1.57E-2 were the flow behavior. considered as the immiscible and miscible limits, respec- The procedure of the relative permeability prediction tively. The proposed default values of the regression param- was carried out in accordance with the flowchart of Figure 1. eters of the equation (6) for linking the capillary number to Hence, the direct interpolation of equations (4) and (5) were relative permeability (A = 2 and Q = 1) were used for curve employed for prediction of kr as a function of two different fitting procedure. Given the velocity ranges utilized in 16 M. Mahdaviara and A. Helalizadeh: Oil & Gas Science and Technology – Rev. IFP Energies nouvelles 75, 24 (2020)

(a)

(b)

Fig. 9. Comparing the accuracy/precision of the new weighting factor of equation (6) (kr–Nc) with five literature published correlations of f1–f5 (Tab. 5) for the condensate phase. (a) The RMSE and (b) R2 entities were employed for determining the fit goodness of each function on five experimental datasets of Table 1. It should be noted that Datasets 4 and 7 were excluded from the analysis because they were not available for the condensate phase. this experiment, the effect of inertia to the relative perme- respectively. Additionally, the values of the regression ability was not observed; therefore, the effect of inertia parameter of Bi were 0.1545 and 0.077 for gas and conden- was ignored. The accuracy and precision of the model for sate phases, respectively, which were situated in the default fitting on the target points were assessed using the range of 0–0.5. statistical parameter of RMSE. Result of the analysis A visual description was represented in Figures 12 demonstrates the exactitude of equation (4) by the and 13 to provide more evaluation regarding the weighting RMSE values of 0.0122 and 0.0418 for gas and condensate functions of equations (4) and (5), respectively. As shown in phases, respectively. Furthermore, the correctness of the Figure 12, the proposed model appropriately follows the equation (5) is also evident by the appropriate RMSE trend of the relative permeability changes with the Nc values of 0.0374 and 0.0049 for gas and condensate phases, and saturation. Furthermore, Figure 13 also demonstrates M. Mahdaviara and A. Helalizadeh: Oil & Gas Science and Technology – Rev. IFP Energies nouvelles 75, 24 (2020) 17

(a)

(b)

Fig. 10. Comparing the accuracy/precision of the new weighting factor of equation (6) (Sr–Nc) with literature published correlations of f1 and f2 (Tab. 5) for the gas phase. (a) The RMSE and (b) R2 entities were employed for determining the fit goodness of each function on six experimental datasets of Table 1. It should be noted that Dataset 1 was excluded from the analysis because it was not reported for this purpose.

that the calculated relative permeability as functions of Nc permeability increases by an increase in the capillary and krg/krc congruously accompany the trends of the target number. Additionally, the influence of the capillary number points. As can be seen, the gas and condensate relative on the gas relative permeability is more conspicuous. This 18 M. Mahdaviara and A. Helalizadeh: Oil & Gas Science and Technology – Rev. IFP Energies nouvelles 75, 24 (2020)

(a)

(b)

Fig. 11. Comparing the accuracy/precision of the new weighting factor of equation (6) (Sr–Nc) with literature published correlations of f1 and f2 (Tab. 5) for the condensate phase. (a) The RMSE and (b) R2 entities were employed for determining the fit goodness of each function on five experimental datasets of Table 1. It should be noted that Datasets 4 and 7 were excluded from the analysis because they were not reported for this purpose. M. Mahdaviara and A. Helalizadeh: Oil & Gas Science and Technology – Rev. IFP Energies nouvelles 75, 24 (2020) 19

(a) (b)

Fig. 12. Investigating the accuracy/precision of the new proposed model for estimation of (a) gas and (b) condensate relative permeabilities as a function of capillary number as well as saturation. The literature published dataset of Blom et al. (1997) was utilized for this purpose.

(a) (b)

Fig. 13. Investigating the accuracy/precision of the new proposed model for estimation of (a) gas and (b) condensate relative permeabilities as a function of capillary number as well as krg/krc. The literature published dataset of Blom et al. (1997) was utilized for this purpose. view is not far from reality when alluding to experimental parameters of RMSE and R2. Results of these evaluations reports of previous studies. It worth pointing out that the are as follows: residual saturation is not a requirement of relative perme- ability estimation using equation (5). Consequently, linking 1. The proposed weighting factor of equation (6) demon- the residual saturation to the capillary number is not strates averagely fair enough consistency with experi- mandatory when calculating the relative permeability as a mental data in various rock/fluid systems. function of krg/krc. 2. A comparison was executed between the developed weighting factor and some literature published corre- lations. The results of this assessment indicated 4 Conclusion that the developed correlations in this study outper- formed the literature published models from the In this paper, a new model was proposed to predict the rel- standpoints of precision and reliability. The recent ative permeability of gas condensate reservoirs. To this end, model benefits from uniformly high enough accuracy the direct interpolation approach was utilized for consider- for different cases. However, the literature models suf- ing both the positive effects of interfacial tension and the fer from high deviations from the target points in low/moderate velocity on the relative permeability. The some cases. unique weighting factor of equation (6) was developed to 3. In order to supply more verification, the fit goodness link the relative permeability as well as residual saturation of the proposed model on a recent dataset that has to the capillary number. For this purpose, regression analy- been excluded from the regression procedure was eval- sis was carried by employing a cumulative number of seven uated. The results of the aforesaid assessment repre- literature published datasets. The accuracy and reliability sented another affirmation for the validity of the of the developed correlations were assessed using statistical developed model. 20 M. Mahdaviara and A. Helalizadeh: Oil & Gas Science and Technology – Rev. IFP Energies nouvelles 75, 24 (2020)

References Dybbs A., Edwards R. (1984) A new look at porous media fluid mechanics – Darcy to turbulent, in: Fundamentals of transport Ahmadi P., Ghandi E., Riazi M., Malayeri M.R. (2019) phenomena in porous media, Springer, Berlin, Germany, pp. Experimental and CFD studies on determination of injection 199–256. and production wells location considering reservoir hetero- Fan L., Harris B.W., Jamaluddin A., Kamath J., Mott R., geneity and capillary number, Oil Gas Sci. Technol. - Rev. Pope G.A., Whitson C.H. (2005) Understanding gas- IFP Energies nouvelles 74,4. condensate reservoirs, Oilfield Rev. 17,4,14–27. Al-Shaidi S.M. (1997) Modelling of gas-condensate flow in Foster W. (1973) A low-tension waterflooding process, J. Pet. reservoir at near wellbore conditions, Heriot-Watt University, Technol. 25, 2, 205–210. Edinburgh, UK. Fulcher R.A. Jr., Ertekin T., Stahl C.D. (1985) Effect of Amaefule J.O., Handy L.L. (1982) The effect of interfacial tensions capillary number and its constituents on two-phase relative on relative oil/water permeabilities of consolidated porous permeability curves, J. Pet. Technol. 37, 2, 249–260. media, Soc. Pet. Eng. J. 22,3,371–381. doi: 10.2118/9783-PA. Guo H., Dou M., Hanqing W., Wang F., Yuanyuan G., Yu Z., App J.F., Burger J.E. (2009) Experimental determination of Yansheng W., Li Y. (2015) Review of capillary number in relative permeabilities for a rich gas/condensate system using chemical enhanced oil recovery, in: Paper Presented at the live fluid, SPE Reserv. Evalu. Eng. 12, 2, 263–269. doi: SPE Kuwait Oil and Gas Show and Conference, 11–14 10.2118/109810-PA. October, Mishref, Kuwait. doi: 10.2118/175172-MS. Asar H., Handy L.L. (1988) Influence of interfacial tension on Haniff M., Ali J. (1990) Relative permeability and low tension gas/oil relative permeability in a gas-condensate system, SPE fluid flow in gas condensate systems, in: Paper Presented at Reserv. Eng. 3, 1, 257–264. the European Petroleum Conference, 21–24 October, The Avraam D.G., Payatakes A.C. (1995) Flow regimes and Hague, The Netherlands. relative permeabilities during steady-state two-phase flow in Henderson G.D., Danesh A., Al-kharusi B., Tehrani D. (2000a) porous media, J. Fluid Mech. 293, 207–236. doi: 10.1017/ Generating reliable gas condensate relative permeability data S0022112095001698. used to develop a correlation with capillary number, J. Pet. Bear J. (2013) Dynamics of fluids in porous media, Courier Sci. Eng. 25,1–2, 79–91. Corporation, Chelmsford, MA. Henderson G., Danesh A., Tehrani D., Al-Kharusi B. (2000b) Betté S., Hartman K., Heinemann R. (1991) Compositional The relative significance of positive coupling and inertial modeling of interfacial tension effects in miscible displacement effects on gas condensate relative permeabilities, in: Paper processes, J. Pet. Sci. Eng. 6,1,1–14. Presented at the SPE Annual Technical Conference and Blom S., Hagoort J. (1998) How to include the capillary number Exhibition, 1–4 October, Dallas, Texas. in gas condensate relative permeability, in: Paper Presented at Henderson G., Danesh A., Tehrani D., Al-Shaidi S., Peden J. the SPE Annual Technical Conference and Exhibition, 27–30 (1988) Measurement and correlation of gas condensate rela- September, New Orleans, Louisiana. tive permeability by the steady-state method, SPE Reserv. Blom S., Hagoort J., Soetekouw D. (1997) Relative permeability Evalu. Eng. 1, 2, 134–140. at near-critical conditions, in: Paper Presented at the SPE Henderson G., Danesh A., Tehrani D., Peden J. (1997) The Annual Technical Conference and Exhibition, 5–8 October, San effect of velocity and interfacial tension on relative perme- Antonio, Texas. ability of gas condensate fluids in the wellbore region, J. Pet. Brooks R., Corey A. (1964) Hydraulic properties of porous Sci. Eng. 17,3–4, 265–273. media, Colorado State University, Fort Collins, CO, Hydro Jamiolahmady M., Danesh A., Tehrani D.H., Sohrabi M. (2006) Paper, 3, 27 p. Variations of gas/condensate relative permeability with pro- Calisgan H., Akin S. (2008) Near critical gas condensate relative duction rate at near-wellbore conditions: A general correlation, permeability of carbonates, Open Pet. Eng. J. 1,1,30–41. SPE Reserv. Evalu. Eng. 9,6,688–697. doi: 10.2118/83960-PA. Chen H., Wilson S., Monger-McClure T. (1995) Determination of Jamiolahmady M., Sohrabi M., Ireland S. (2008) Gas- relative permeability and recovery for North Sea gas condensate condensate relative permeabilities in propped fracture reservoirs, in: Paper Presented at the SPE Annual Technical porous media: Coupling vs. Inertia, in: Paper Presented at Conference and Exhibition, 22–25 October, Dallas, Texas. the SPE Annual Technical Conference and Exhibition, 21–24 Chukwudeme E.A., Fjelde I., Abeysinghe K.P., Lohne A. (2014) September, Denver, Colorado. Effect of interfacial tension on water/oil relative permeability Jamiolahmady M., Sohrabi M., Ireland S., Ghahri P. (2009) A on the basis of history matching to coreflood data, SPE generalized correlation for predicting gas–condensate relative Reserv. Evalu. Eng. 17,1,37–48. permeability at near wellbore conditions, J. Pet. Sci. Eng. 66, Coats K.H. (1980) An equation of state compositional model, 3–4, 98–110. Soc. Pet. Eng. J. 20, 5, 363–376. doi: 10.2118/8284-PA. Johannesen E.B., Graue A. (2007) Mobilization of remaining oil – Delshad M., Bhuyan D., Pope G., Lake L. (1986) Effect of emphasis on capillary number and wettability, in: Paper capillary number on the residual saturation of a three-phase Presented at the International Oil Conference and Exhibition in micellar solution, in: Paper Presented at the SPE Enhanced Mexico, 27–30 June, Veracruz, Mexico.doi:10.2118/108724-MS. Oil Recovery Symposium, 20–23 April, Tulsa, Oklahoma. Kalla S., Leonardi S.A., Berry D.W., Poore L.D., Sahoo H., Delshad M., Najafabadi N.F., Anderson G., Pope G.A., Kudva R.A., Braun E.M. (2014) Factors that affect gas- Sepehrnoori K. (2009) Modeling wettability alteration by condensate relative permeability, in: Paper Presented at the surfactants in naturally fractured reservoirs, SPE Reserv. IPTC 2014 International Petroleum Technology Conference, Evalu. Eng. 12, 3, 361–370. doi: 10.2118/100081-PA. 19–22 January, Doha, Qatar. Du Plessis J.P., Woudberg S. (2008) Pore-scale derivation of the Lenormand R., Touboul E., Zarcone C. (1988) Numerical models Ergun equation to enhance its adaptability and generaliza- and experiments on immiscible displacements in porous tion, Chem. Eng. Sci. 63, 9, 2576–2586. media, J. Fluid Mech. 189, 165–187. M. Mahdaviara and A. Helalizadeh: Oil & Gas Science and Technology – Rev. IFP Energies nouvelles 75, 24 (2020) 21

Lomeland F., Ebeltoft E., Thomas W.H. (2005) A new versatile Tsakiroglou C.D. (2019) The correlation of the steady-state gas/ relative permeability correlation, in: Paper Presented at the water relative permeabilities of porous media with gas and International Symposium of the Society of Core Analysts, water capillary numbers, Oil Gas Sci. Technol. - Rev. IFP 21–25 August, Toronto, Canada. Energies nouvelles 74, 45. Longeron D. (1980) Influence of very low interfacial tensions on Tsakiroglou C.D., Aggelopoulos C.A., Terzi K., Avraam D.G., relative permeability, Soc. Pet. Eng. J. 20, 5, 391–401. Valavanides M.S. (2015) Steady-state two-phase relative Mott R., Cable A., Spearing M. (1999) A new method of permeability functions of porous media: A revisit, Int. J. measuring relative permeabilities for calculating gas-conden- Multiph. Flow 73,34–42. sate well deliverability, in: Paper Presented at the SPE Annual Tsakiroglou C.D., Avraam D.G., Payatakes A.C. (2007) Tran- Technical Conference and Exhibition, 3–6 October, Houston, sient and steady-state relative permeabilities from two-phase Texas. flow experiments in planar pore networks, Adv. Water Res. Nghiem L.X., Fong D., Aziz K. (1981) Compositional modeling 30, 9, 1981–1992. doi: 10.1016/j.advwatres.2007.04.002. with an equation of state (includes associated papers 10894 Urumović K., Urumović K. Sr. (2016) The referential grain size and 10903), Soc. Pet. Eng. J. 21, 6, 687–698. and effective porosity in the Kozeny-Carman model, Hydrol. Payatakes A. (1982) Dynamics of oil ganglia during immiscible Earth Syst. Sci. 20, 5, 1669–1680. displacement in water-wet porous media, Annu. Rev. Fluid van Lopik J.H., Snoeijers R., van Dooren T.C., Raoof A., Mech. 14, 1, 365–393. Schotting R.J. (2017) The effect of grain size distribution on Pope G.A., Wu W., Narayanaswamy G., Delshad M., nonlinear flow behavior in sandy porous media, Transp. Sharma M.M., Wang P. (2000) Modeling relative permeability Porous Media 120,1,37–66. effects in gas-condensate reservoirs with a new trapping Whitson C.H., Fevang Ø. (1997) Generalized pseudopressure model, SPE Reserv. Evalu. Eng. 3, 2, 171–178. doi: 10.2118/ well treatment in reservoir simulation, in: Paper Presented at 62497-PA. the Proc. IBC Conference on Optimisation of Gas Condensate Stone H.L. (1973) Estimation of three-phase relative permeabil- Fields,26–27 June, Aberdeen, United Kingdom. ity and residual oil data, J. Can. Pet. Technol. 12,4,53–61. Whitson C.H., Fevang Ø., Sævareid A. (1999) Gas condensate Timur A. (1968) An investigation of permeability, porosity, and relative permeability for well calculations, in: Paper Presented residual water saturation relationships for sandstone reser- at the SPE Annual Technical Conference and Exhibition, 3–6 voirs, Log Anal. 9,4,3–5. October, Houston, Texas.